Exact and Approximate Expansions with Pure Gaussian Wave Packets

j,k and aligned at an angle of 2⇡k 2j radians. Like other parabolic systems, such as curvelets and shearlets, these wavepackets are particularly useful in geophysics (see for example [7]). In this paper, we consider expansions and approximate expansions of L2(R2) functions in terms of these packets using Gaussian windows. As with Morlet’s wavelets, one can circumvent the problem of the lack of vanishing moments by considering linear combinations of modulated Gaussians. As a fundamental example we consider windows of the form,